1. Field of the Invention
The present invention relates to a chaos feedback system, more particularly to a chaos feedback system which is able to determine whether an input signal is obtained from a random noise or from a meaningful information by analyzing qualitative features of the strange attractor and feed the analyzed result back to a chaotic system so that the chaotic system outputs a desired state.
2. Description of the Prior Art
Recently, there have been active studies for seeking a process for estimating the future movements of the nature phenomena, such as the flow of water, air, and blood etc., the movement may be predetermined by the mathematically expressed regularity of the movements gained.
Dynamics system can be defined as a system in which its states are varying with respect to time.
The dynamics system is called a stable system when the steady state solution remains in one point, which is in turn called an equilibrium point. When the attractor of the system makes a closed loop, the system is called a periodic system. When the attractor has a shape of doughnut, it is called a quasi-periodic system.
The procedure for obtaining the attractor of the dynamics system will be described as follows.
Generally, an n-th order dynamics system have n state equations, and the state equations indicate the variation ratio of the states of the dynamics system depending on the variation of time as expressed in equation (1). ##EQU1## wherein, f: R.sup.n .fwdarw.R.sup.n represents a nonlinear mapping, and x1, x2, . . . , xn represent states respectively.
Hereinafter, a pendulum motion will be described as an elemental example of the dynamics system, the pendulum motion is expressed by 2 state equations in that the pendulum motion is a second order dynamics system as follows. EQU dx1/dt=f(x1, x2) EQU dx2/dt=f(x1, x2)
The solutions of the above state equations consist of a transient solution and a steady state solution. The steady state solution can be expressed in a state space, in which each state variable makes an axis of the state space, so as to express the steady state solution entirely.
Namely, the steady state solution at a give time t can be expressed as a point in the state space. A set of the points presented in the state space is called an attractor of dynamics system.
If the given dynamics system has a finite state, i.e., a finite n-th order dynamics system, the system has a four forms of the attractor. The dynamics systems are classified into four types according to the types of attractor of the dynamics system.
Namely, the steady linear system which is the most simple dynamics system has one point attractor in the state space, which is called an equilibrium point. Also, the dynamics system having the steady state solution, and the solution being a periodic solution, has a closed loop-shaped attractor in the state space, which is then called a limit cycle.
And, the dynamics system having k-th order subharmonic solution, which has k periods, has a doughnut-shaped attractor. The doughnut is called a torus.
The attractor except those of the above-mentioned dynamics systems is a strange attractor, and this type of dynamics system is called a chaos system.
Namely, the chaos system refers to a system having a strange attractor in the state space, with the exception of the one point attractor, the limit cycle, and the torus.
As mentioned above, the attractor may be constructed from the state equation which represents the state of the movements of the nature phenomena. In that case, all the n state equations are known in the n-th order dynamics system, the attractor may be constructed easily.
In fact, however, it is practically impossible to access the whole n state equations let alone state variables, in a given n-th order dynamics system. Accordingly, the endeavor has been devoted to construct the attractor of n-th order dynamics system from only one state variable.
Namely, when an attractor is obtained from the steady state solution of a given state variable, the attractor may be presented in the state space.
Since the desired attractor may not be gained in the state space, an embedding space should be introduced.
As described above, the constructing of the attractor of the n-th dynamics system from a given state variable is called an attractor reconstruction. The attractor reconstruction plays an important role among the researchers who are dependent upon the experiments.
The attractor reconstruction has been proposed by Tarkens in the mid of 1980's.
And, the trace time is divided into the same periods, and the corresponding state value of the divided time is presented as a vector, g(t). The vector g(t) is satisfied with the following equation: EQU g(t)={y(t), y(t+.tau.), . . . , y(t+n.tau.)}
wherein, y(t) represents a state value, .tau. is a delay time divided into the same periods, and n+1 is an embedding dimension.
If the delay time and the embedding dimension are fixed, the vector is expressed as one point. And the delay time and the embedding dimension are altered, then the vector draws a trace in the embedding space.
The vector trace in the given n-th order embedding space may not be exactly the same as the trace of the attractor of the given dynamics system, but the vector trace has relation with the trace of the original dynamics system in the qualitative viewpoint (pattern face).
However, it needs to determine whether the attractor is constructed from a random noise or the meaningful information from chaos system.
There are two kinds of methods for analyzing the reconstructed attractor, one being to analyze the qualitative feature of the reconstructed attractor, which is called a qualitative method, and the other being to analyze the degree of the pattern such as a slope of the attractor, which is called a quantitative method.
In the case that the reconstructed attractor has a pattern of the equilibrium point, the limit cycle, or the torus, it is possible to analyze the attractor only by use of the qualitative method. Where the strange attractor is constructed by the reconstruction, it is impossible to determine whether the attractor is constructed from a noise or a meaningful information only by use of the qualitative process. In this case, therefore, the strange attractor is analyzed by analyzing the quantitative feature of the attractor.
There are various methods in analyzing the quantitative feature of the reconstructed strange attractor such as the procedure for calculating the capacity of the strange attractor, the procedure for gaining an information dimension, and the procedure for gaining a correlation dimension and the like.
The process for calculating the capacity of the reconstructed strange attractor will be described hereinafter.
Assuming that the reconstructed strange attractor is covered with volume elements with a radius r and a shape of, for example, a sphere or a hexahedron, and that the number of the volume elements necessary to cover entirely the attractor is N(r), the relation N(r)=kr.sup.0 is satisfied.
In a case that the radius r is reduced enough, then n(r) is solved with regard to the D, the capacity of attractor D.sub.cap satisfies the following equation: ##EQU2##
The process for gaining the capacity of attractor is carried out by using the space, but it does not use the information accompanying the state variation of the given dynamics system.
Namely, the information dimension employs the following equation in analyzing the quantitative feature of the attractor which is reconstructed by using the information accompanying the state variation of the dynamics system. ##EQU3##
The P.sub.i represents a probability in which the trace enters the n-th volume element, and I(r) represents an entropy of the given dynamics system.
In the meanwhile, the most convenient procedure for analyzing the quantitative feature of the reconstructed attractor is to gain the correlation dimension. This method will be described with detail hereinafter.
First, it is gained that the number of states present in a circle having a radius Ri which corresponds to the distance between two states Xi and Xj. The gained number of the states is divided by the square number of the state value (N) of the attractor, that is, N.sup.2. As the whole number (N) is approached to the infinite, the correlation sum C(R) of the state Xi satisfying the following equation (1) can be obtained: EQU C(R)=lim 1/N.sup.2 {the number of states (Xi, Xj) such that .vertline..vertline.X.sub.i -X.sub.j .vertline..vertline.&lt;R}(1).
With the above calculated correlation sum C(R), the correlation dimension Dc can be calculated by using the following equation: ##EQU4##
The correlation dimension Dc stands for the slope of the linear part of a graph of the correlation sum C(R) which is calculated by the equation (1).
Namely, the graph of the correlation sum C(R) calculated by the equation (1) is plotted in a form convergent toward a certain value. However, the attractor which is constructed by the noise is plotted in a divergent form.
And, the correlation dimension Dc of the attractor is obtained from the slope of the linear part of the graph which is gained by the equation (1).
As described above, in analyzing the quantitative feature of the reconstructed strange attractor, a circuit seeking the correlation dimension is called a chaos processor.
In the prior art, the chaos processor requires a great large amount of calculation works, in that the correlation sum is to obtained from the whole states X1, X2, . . . , Xn. Namely, in order to get the correlation sum, the calculation which is equivalent to the square number of the whole states N.sup.2 (N.sup.2 =N(N-1)/2) is needed. The more the number of state values, the real time processing for the calculation becomes more hardly achieved by use of a common computer. It is also impossible to use the result of chaos system in the real time.